The collective plasmon modes of a doped or modulated doped semiconductor superlattice are investigated via a calculation of the wave-vector frequency-dependent dielectric constant. Explicit account is taken of the quantum well subband structure; including finite wave functions and intersubband transitions. Superlattice plasmons with dispersion quite different from bulk plasmons are found. Results for the interaction of light with these superlattice plasmons will also be presented. A new intersubband plasmon mode involving the coupling of the charge depolarization response into modes propagating along the superlattice axis is predicted. These collective excitations are also investigated in the classical regime where the electron deBroglie wavelength is much less than the well size. Here, we consider a superlattice of metal slabs, each capable of supporting symmetric and antisymmetric surface plasmon modes which couple into corresponding plasmon bands along the superlattice axis. The relationship between these modes and the ground superlattice plasmon and intersubband plasmon is derived.